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  2. Rigid transformation - Wikipedia

    en.wikipedia.org/wiki/Rigid_transformation

    In dimension at most three, any improper rigid transformation can be decomposed into an improper rotation followed by a translation, or into a sequence of reflections. Any object will keep the same shape and size after a proper rigid transformation. All rigid transformations are examples of affine transformations. The set of all (proper and ...

  3. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    Alibi and alias transformations are also known as active and passive transformations, respectively. Pre-multiplication or post-multiplication The same point P can be represented either by a column vector v or a row vector w. Rotation matrices can either pre-multiply column vectors (Rv), or post-multiply row vectors (wR).

  4. Rotation of axes in two dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_of_axes_in_two...

    The equations defining the transformation in two dimensions, which rotates the xy axes counterclockwise through an angle into the x′y′ axes, are derived as follows. In the xy system, let the point P have polar coordinates ( r , α ) {\displaystyle (r,\alpha )} .

  5. Affine transformation - Wikipedia

    en.wikipedia.org/wiki/Affine_transformation

    Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...

  6. Euclidean group - Wikipedia

    en.wikipedia.org/wiki/Euclidean_group

    The position and orientation of the body at any later time t will be described by the transformation f(t). Since f(0) = I is in E + (3), the same must be true of f(t) for any later time. For that reason, the direct Euclidean isometries are also called "rigid motions".

  7. Rotation formalisms in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_formalisms_in...

    Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system). The basic problem is to specify the orientation of these three unit vectors, and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.

  8. Similarity (geometry) - Wikipedia

    en.wikipedia.org/wiki/Similarity_(geometry)

    A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection f from the space onto itself that multiplies all distances by the same positive real number r, so that for any two points x and y we have ((), ()) = (,), where d(x,y) is the Euclidean distance from x to y. [16]

  9. List of common coordinate transformations - Wikipedia

    en.wikipedia.org/wiki/List_of_common_coordinate...

    Note: solving for ′ returns the resultant angle in the first quadrant (< <). To find , one must refer to the original Cartesian coordinate, determine the quadrant in which lies (for example, (3,−3) [Cartesian] lies in QIV), then use the following to solve for :